Legendrian contact homology in Seifert fibered spaces

Abstract

We define a differential graded algebra associated to a Legendrian knot in a Seifert fibered space with a transverse contact structure. The new feature of this construction is the existence of orbifold points in the Reeb orbit space of the contact manifold. These orbifold points are images of the exceptional fibers of the Seifert fibered manifold, and they play a key role in the definitions of the differential and the grading, as well as in the proof of invariance. We apply the invariant to distinguish Legendrian knots whose homology is torsion and whose underlying topological knot types are isotopic; such examples exist in any sufficiently complicated contact Seifert fibered space.